Circular railway

The railway connects points A, B, and C in a circular arc, whose distances are | AB | = 30 km, AC = 95 km, and BC | = 70 km. How long will the track be from A to C?

Correct answer:

b₁ =  103.1095 km

Step-by-step explanation:

$$\begin{gathered} c=30\text{km} \\ b=95\text{km} \\ a=70\text{km} \\ s=\frac{a+b+c}{2}=\frac{70+95+30}{2}=\frac{195}{2}=97.5\text{km} \\ S=\sqrt{s\cdot (s-a)\cdot (s-b)\cdot (s-c)}=\sqrt{97.5\cdot (97.5-70)\cdot (97.5-95)\cdot (97.5-30)}\doteq 672.6522{\text{km}}^2 \\ r=S/s=672.6522/97.5\doteq 6.899\text{km} \\ R=\frac{a\cdot b\cdot c}{4\cdot r\cdot s}=\frac{70\cdot 95\cdot 30}{4\cdot 6.899\cdot 97.5}\doteq 74.1468\text{km} \\ \Delta R-R-b \\ {b}^2=R^2+R^2-2R{}^2\cos \theta \\ \theta =\arccos (\frac{2\cdot R^2-b^2}{2\cdot R^2})=\arccos (\frac{2\cdot 74.1468^2-95^2}{2\cdot 74.1468^2})\doteq 1.3906\text{rad} \\ {b}_1=\theta \cdot R=1.3906\cdot 74.1468=103.1095\text{km} \end{gathered}$$

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